Algebraic Topology

Algebraic Topology is a field of mathematics combining two distinct areas: Algebra and Topology. Its main goal is to find and develop certain properties of topological spaces (in two dimensions, these are essentially surfaces) that remain invariant under certain types of transformations to be able to classify surfaces. To understand what all this means and why algebra is needed, the fields of Topology and Algebra need to be elaborated on.

An important transformation between two topological surfaces is the homeomorphism. A homeomorphism f is a function from one topological space X to a topological space Y that satisfies certain properties: every point in X is mapped to one and only one point in Y, for every point in Y, a point in X can be found that is mapped to the point in Y, every open set (in two dimensions, can be thought of as the set of points taken up by a surface minus the points on the surface's boundary) in Y is mapped from an open set in X, and there exists some function f^(-1) such that the composition of f and f^(-1) is the identity function, and that for f^(-1), every open set in X is mapped from an open set in Y. The first condition says f is injective or one-to-one and the second condition says that f is surjective or onto. The third and fourth conditions state that f is continuous and has a continuous inverse. Homeomorphisms are studied because they are very special and behave very nicely-they preserve open sets, which also can be thought of as preserving neighborhoods-two points next to each other will always be mapped to two points next to each other. This is essentially what Topology is all about, because Topology deals not with distance as Geometry does but with "nearness". Topology can be thought of as a general form of Geometry, where distance is abstracted to "nearness". In Topology, if there exists a homeomorphism between two spaces, then the spaces can be thought of as equivalent. In this way, a donut and a coffee cup can be thought of as equivalent. Now the question is asked, how can one determine whether two spaces are homeomorphic to each other? This is where a diversion into Algebra is needed.

The Algebra that is referred to here is not high school algebra-it does not deal directly with such topics such as quadratic equations or conic sections. Rather, it is much more general, and is called "Abstract Algebra". There is a certain field in Abstract Algebra that can be applied to great effect in Topology; this topic is called "Group Theory" and deals with mathematical objects called groups. A group G is simply a set of elements endowed with some operation * with the following properties under the operation.

1.  For all x,y in G, x*y is an element in G
2. There exists an element e such that for all x in G such that x*e=e*x=x
3. For all x,y,z in G, (x*y)*z=x*(y*z)
4. For all x in G, there exists an element y in G such that x*y=y*x=e

  The first condition is G is closed under *, the second is the existence of an identity element, the third is transitivity, and the fourth is the existence of an inverse. Some examples of groups are the integers under addition, (Z under +), the reals minus 0 under multiplication ( R\{0} under x), and n dimensional space under vector addition ( R^n under +). In all these examples, the elements are numbers or points, but in general, the elements could be any element, even something as unspecific as the letter A. An interesting application of Group Theory is solving the Rubik's Cube, but now back to Topology; how can groups be applied to the study of Topology?
 

When viewed a particular way, surfaces can be assigned a group. Given two curves on a surface who have the same endpoints, a homotopy exists between the two if and only if there is a continuous function F(s,t) such that at t=0, F maps to one of the curves, while at t=1, F maps to the other curve. Essentially, two curves are homotopic if there exists  a continuous "deformation" from one curve to the other. It turns out that on surfaces, the set of sets of curves that are homotopic to one another but not to other sets of curves forms a group under the operation of path class multiplication, i.e. path concatenation. This is called the fundamental group of a surface. Surfaces with different fundamental groups are never homeomorphic to each other (the converse is not always true), and homeomorphisms between two surfaces preserve the surfaces' fundamental groups. Therefore the fundamental group of a surface is an invariant, and can be used to classify surfaces. Some examples of fundamental groups are that the circle has a fundamental group of Z (the integers), the two dimensional surface of the sphere has a trivial fundamental group (all curves are homotopic to each other, and therefore there is only one element in the group, which must be the identity element), and the torus (donut), which has a fundamental group of ZxZ (all ordered pairs of integers (a,b)).

The fundamental group of surfaces can be generalized to the fundamental group on manifolds (n dimensional surfaces). In n dimensions, the fundamental group would simply be the set of sets of (n-1) manifolds who are homotopic to each other but not to other sets of (n-1) manifolds. These groups are called higher homotopy groups.

By synthesizing two distinct fields of mathematics, mathematicians have successfully advanced the study of types and classes of surfaces and manifolds.